// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_GENERIC_PACKET_MATH_H
#define EIGEN_GENERIC_PACKET_MATH_H

namespace Eigen {

namespace internal {

    /** \internal
  * \file GenericPacketMath.h
  *
  * Default implementation for types not supported by the vectorization.
  * In practice these functions are provided to make easier the writing
  * of generic vectorized code.
  */

#ifndef EIGEN_DEBUG_ALIGNED_LOAD
#define EIGEN_DEBUG_ALIGNED_LOAD
#endif

#ifndef EIGEN_DEBUG_UNALIGNED_LOAD
#define EIGEN_DEBUG_UNALIGNED_LOAD
#endif

#ifndef EIGEN_DEBUG_ALIGNED_STORE
#define EIGEN_DEBUG_ALIGNED_STORE
#endif

#ifndef EIGEN_DEBUG_UNALIGNED_STORE
#define EIGEN_DEBUG_UNALIGNED_STORE
#endif

    struct default_packet_traits
    {
        enum
        {
            HasHalfPacket = 0,

            HasAdd = 1,
            HasSub = 1,
            HasShift = 1,
            HasMul = 1,
            HasNegate = 1,
            HasAbs = 1,
            HasArg = 0,
            HasAbs2 = 1,
            HasAbsDiff = 0,
            HasMin = 1,
            HasMax = 1,
            HasConj = 1,
            HasSetLinear = 1,
            HasBlend = 0,
            // This flag is used to indicate whether packet comparison is supported.
            // pcmp_eq, pcmp_lt and pcmp_le should be defined for it to be true.
            HasCmp = 0,

            HasDiv = 0,
            HasSqrt = 0,
            HasRsqrt = 0,
            HasExp = 0,
            HasExpm1 = 0,
            HasLog = 0,
            HasLog1p = 0,
            HasLog10 = 0,
            HasPow = 0,

            HasSin = 0,
            HasCos = 0,
            HasTan = 0,
            HasASin = 0,
            HasACos = 0,
            HasATan = 0,
            HasSinh = 0,
            HasCosh = 0,
            HasTanh = 0,
            HasLGamma = 0,
            HasDiGamma = 0,
            HasZeta = 0,
            HasPolygamma = 0,
            HasErf = 0,
            HasErfc = 0,
            HasNdtri = 0,
            HasBessel = 0,
            HasIGamma = 0,
            HasIGammaDerA = 0,
            HasGammaSampleDerAlpha = 0,
            HasIGammac = 0,
            HasBetaInc = 0,

            HasRound = 0,
            HasRint = 0,
            HasFloor = 0,
            HasCeil = 0,
            HasSign = 0
        };
    };

    template <typename T> struct packet_traits : default_packet_traits
    {
        typedef T type;
        typedef T half;
        enum
        {
            Vectorizable = 0,
            size = 1,
            AlignedOnScalar = 0,
            HasHalfPacket = 0
        };
        enum
        {
            HasAdd = 0,
            HasSub = 0,
            HasMul = 0,
            HasNegate = 0,
            HasAbs = 0,
            HasAbs2 = 0,
            HasMin = 0,
            HasMax = 0,
            HasConj = 0,
            HasSetLinear = 0
        };
    };

    template <typename T> struct packet_traits<const T> : packet_traits<T>
    {
    };

    template <typename T> struct unpacket_traits
    {
        typedef T type;
        typedef T half;
        enum
        {
            size = 1,
            alignment = 1,
            vectorizable = false,
            masked_load_available = false,
            masked_store_available = false
        };
    };

    template <typename T> struct unpacket_traits<const T> : unpacket_traits<T>
    {
    };

    template <typename Src, typename Tgt> struct type_casting_traits
    {
        enum
        {
            VectorizedCast = 0,
            SrcCoeffRatio = 1,
            TgtCoeffRatio = 1
        };
    };

    /** \internal Wrapper to ensure that multiple packet types can map to the same
    same underlying vector type. */
    template <typename T, int unique_id = 0> struct eigen_packet_wrapper
    {
        EIGEN_ALWAYS_INLINE operator T&() { return m_val; }
        EIGEN_ALWAYS_INLINE operator const T&() const { return m_val; }
        EIGEN_ALWAYS_INLINE eigen_packet_wrapper() {}
        EIGEN_ALWAYS_INLINE eigen_packet_wrapper(const T& v) : m_val(v) {}
        EIGEN_ALWAYS_INLINE eigen_packet_wrapper& operator=(const T& v)
        {
            m_val = v;
            return *this;
        }

        T m_val;
    };

    /** \internal A convenience utility for determining if the type is a scalar.
 * This is used to enable some generic packet implementations.
 */
    template <typename Packet> struct is_scalar
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        enum
        {
            value = internal::is_same<Packet, Scalar>::value
        };
    };

    /** \internal \returns static_cast<TgtType>(a) (coeff-wise) */
    template <typename SrcPacket, typename TgtPacket> EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a) { return static_cast<TgtPacket>(a); }
    template <typename SrcPacket, typename TgtPacket> EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a, const SrcPacket& /*b*/)
    {
        return static_cast<TgtPacket>(a);
    }
    template <typename SrcPacket, typename TgtPacket>
    EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a, const SrcPacket& /*b*/, const SrcPacket& /*c*/, const SrcPacket& /*d*/)
    {
        return static_cast<TgtPacket>(a);
    }
    template <typename SrcPacket, typename TgtPacket>
    EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a,
                                             const SrcPacket& /*b*/,
                                             const SrcPacket& /*c*/,
                                             const SrcPacket& /*d*/,
                                             const SrcPacket& /*e*/,
                                             const SrcPacket& /*f*/,
                                             const SrcPacket& /*g*/,
                                             const SrcPacket& /*h*/)
    {
        return static_cast<TgtPacket>(a);
    }

    /** \internal \returns reinterpret_cast<Target>(a) */
    template <typename Target, typename Packet>
    EIGEN_DEVICE_FUNC inline Target preinterpret(const Packet& a); /* { return reinterpret_cast<const Target&>(a); } */

    /** \internal \returns a + b (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet padd(const Packet& a, const Packet& b) { return a + b; }
    // Avoid compiler warning for boolean algebra.
    template <> EIGEN_DEVICE_FUNC inline bool padd(const bool& a, const bool& b) { return a || b; }

    /** \internal \returns a - b (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet psub(const Packet& a, const Packet& b) { return a - b; }

    /** \internal \returns -a (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pnegate(const Packet& a) { return -a; }

    template <> EIGEN_DEVICE_FUNC inline bool pnegate(const bool& a) { return !a; }

    /** \internal \returns conj(a) (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pconj(const Packet& a) { return numext::conj(a); }

    /** \internal \returns a * b (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pmul(const Packet& a, const Packet& b) { return a * b; }
    // Avoid compiler warning for boolean algebra.
    template <> EIGEN_DEVICE_FUNC inline bool pmul(const bool& a, const bool& b) { return a && b; }

    /** \internal \returns a / b (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pdiv(const Packet& a, const Packet& b) { return a / b; }

    // In the generic case, memset to all one bits.
    template <typename Packet, typename EnableIf = void> struct ptrue_impl
    {
        static EIGEN_DEVICE_FUNC inline Packet run(const Packet& /*a*/)
        {
            Packet b;
            memset(static_cast<void*>(&b), 0xff, sizeof(Packet));
            return b;
        }
    };

    // For non-trivial scalars, set to Scalar(1) (i.e. a non-zero value).
    // Although this is technically not a valid bitmask, the scalar path for pselect
    // uses a comparison to zero, so this should still work in most cases. We don't
    // have another option, since the scalar type requires initialization.
    template <typename T> struct ptrue_impl<T, typename internal::enable_if<is_scalar<T>::value && NumTraits<T>::RequireInitialization>::type>
    {
        static EIGEN_DEVICE_FUNC inline T run(const T& /*a*/) { return T(1); }
    };

    /** \internal \returns one bits. */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet ptrue(const Packet& a) { return ptrue_impl<Packet>::run(a); }

    // In the general case, memset to zero.
    template <typename Packet, typename EnableIf = void> struct pzero_impl
    {
        static EIGEN_DEVICE_FUNC inline Packet run(const Packet& /*a*/)
        {
            Packet b;
            memset(static_cast<void*>(&b), 0x00, sizeof(Packet));
            return b;
        }
    };

    // For scalars, explicitly set to Scalar(0), since the underlying representation
    // for zero may not consist of all-zero bits.
    template <typename T> struct pzero_impl<T, typename internal::enable_if<is_scalar<T>::value>::type>
    {
        static EIGEN_DEVICE_FUNC inline T run(const T& /*a*/) { return T(0); }
    };

    /** \internal \returns packet of zeros */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pzero(const Packet& a) { return pzero_impl<Packet>::run(a); }

    /** \internal \returns a <= b as a bit mask */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pcmp_le(const Packet& a, const Packet& b) { return a <= b ? ptrue(a) : pzero(a); }

    /** \internal \returns a < b as a bit mask */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pcmp_lt(const Packet& a, const Packet& b) { return a < b ? ptrue(a) : pzero(a); }

    /** \internal \returns a == b as a bit mask */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pcmp_eq(const Packet& a, const Packet& b) { return a == b ? ptrue(a) : pzero(a); }

    /** \internal \returns a < b or a==NaN or b==NaN as a bit mask */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pcmp_lt_or_nan(const Packet& a, const Packet& b) { return a >= b ? pzero(a) : ptrue(a); }

    template <typename T> struct bit_and
    {
        EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a, const T& b) const { return a & b; }
    };

    template <typename T> struct bit_or
    {
        EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a, const T& b) const { return a | b; }
    };

    template <typename T> struct bit_xor
    {
        EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a, const T& b) const { return a ^ b; }
    };

    template <typename T> struct bit_not
    {
        EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a) const { return ~a; }
    };

    // Use operators &, |, ^, ~.
    template <typename T> struct operator_bitwise_helper
    {
        EIGEN_DEVICE_FUNC static inline T bitwise_and(const T& a, const T& b) { return bit_and<T>()(a, b); }
        EIGEN_DEVICE_FUNC static inline T bitwise_or(const T& a, const T& b) { return bit_or<T>()(a, b); }
        EIGEN_DEVICE_FUNC static inline T bitwise_xor(const T& a, const T& b) { return bit_xor<T>()(a, b); }
        EIGEN_DEVICE_FUNC static inline T bitwise_not(const T& a) { return bit_not<T>()(a); }
    };

    // Apply binary operations byte-by-byte
    template <typename T> struct bytewise_bitwise_helper
    {
        EIGEN_DEVICE_FUNC static inline T bitwise_and(const T& a, const T& b) { return binary(a, b, bit_and<unsigned char>()); }
        EIGEN_DEVICE_FUNC static inline T bitwise_or(const T& a, const T& b) { return binary(a, b, bit_or<unsigned char>()); }
        EIGEN_DEVICE_FUNC static inline T bitwise_xor(const T& a, const T& b) { return binary(a, b, bit_xor<unsigned char>()); }
        EIGEN_DEVICE_FUNC static inline T bitwise_not(const T& a) { return unary(a, bit_not<unsigned char>()); }

    private:
        template <typename Op> EIGEN_DEVICE_FUNC static inline T unary(const T& a, Op op)
        {
            const unsigned char* a_ptr = reinterpret_cast<const unsigned char*>(&a);
            T c;
            unsigned char* c_ptr = reinterpret_cast<unsigned char*>(&c);
            for (size_t i = 0; i < sizeof(T); ++i) { *c_ptr++ = op(*a_ptr++); }
            return c;
        }

        template <typename Op> EIGEN_DEVICE_FUNC static inline T binary(const T& a, const T& b, Op op)
        {
            const unsigned char* a_ptr = reinterpret_cast<const unsigned char*>(&a);
            const unsigned char* b_ptr = reinterpret_cast<const unsigned char*>(&b);
            T c;
            unsigned char* c_ptr = reinterpret_cast<unsigned char*>(&c);
            for (size_t i = 0; i < sizeof(T); ++i) { *c_ptr++ = op(*a_ptr++, *b_ptr++); }
            return c;
        }
    };

    // In the general case, use byte-by-byte manipulation.
    template <typename T, typename EnableIf = void> struct bitwise_helper : public bytewise_bitwise_helper<T>
    {
    };

    // For integers or non-trivial scalars, use binary operators.
    template <typename T>
    struct bitwise_helper<T, typename internal::enable_if<is_scalar<T>::value && (NumTraits<T>::IsInteger || NumTraits<T>::RequireInitialization)>::type>
        : public operator_bitwise_helper<T>
    {
    };

    /** \internal \returns the bitwise and of \a a and \a b */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pand(const Packet& a, const Packet& b) { return bitwise_helper<Packet>::bitwise_and(a, b); }

    /** \internal \returns the bitwise or of \a a and \a b */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet por(const Packet& a, const Packet& b) { return bitwise_helper<Packet>::bitwise_or(a, b); }

    /** \internal \returns the bitwise xor of \a a and \a b */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pxor(const Packet& a, const Packet& b) { return bitwise_helper<Packet>::bitwise_xor(a, b); }

    /** \internal \returns the bitwise not of \a a */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pnot(const Packet& a) { return bitwise_helper<Packet>::bitwise_not(a); }

    /** \internal \returns the bitwise and of \a a and not \a b */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pandnot(const Packet& a, const Packet& b) { return pand(a, pnot(b)); }

    // In the general case, use bitwise select.
    template <typename Packet, typename EnableIf = void> struct pselect_impl
    {
        static EIGEN_DEVICE_FUNC inline Packet run(const Packet& mask, const Packet& a, const Packet& b) { return por(pand(a, mask), pandnot(b, mask)); }
    };

    // For scalars, use ternary select.
    template <typename Packet> struct pselect_impl<Packet, typename internal::enable_if<is_scalar<Packet>::value>::type>
    {
        static EIGEN_DEVICE_FUNC inline Packet run(const Packet& mask, const Packet& a, const Packet& b)
        {
            return numext::equal_strict(mask, Packet(0)) ? b : a;
        }
    };

    /** \internal \returns \a or \b for each field in packet according to \mask */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pselect(const Packet& mask, const Packet& a, const Packet& b)
    {
        return pselect_impl<Packet>::run(mask, a, b);
    }

    template <> EIGEN_DEVICE_FUNC inline bool pselect<bool>(const bool& cond, const bool& a, const bool& b) { return cond ? a : b; }

    /** \internal \returns the min or of \a a and \a b (coeff-wise)
    If either \a a or \a b are NaN, the result is implementation defined. */
    template <int NaNPropagation> struct pminmax_impl
    {
        template <typename Packet, typename Op> static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a, const Packet& b, Op op) { return op(a, b); }
    };

    /** \internal \returns the min or max of \a a and \a b (coeff-wise)
    If either \a a or \a b are NaN, NaN is returned. */
    template <> struct pminmax_impl<PropagateNaN>
    {
        template <typename Packet, typename Op> static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a, const Packet& b, Op op)
        {
            Packet not_nan_mask_a = pcmp_eq(a, a);
            Packet not_nan_mask_b = pcmp_eq(b, b);
            return pselect(not_nan_mask_a, pselect(not_nan_mask_b, op(a, b), b), a);
        }
    };

    /** \internal \returns the min or max of \a a and \a b (coeff-wise)
    If both \a a and \a b are NaN, NaN is returned.
    Equivalent to std::fmin(a, b).  */
    template <> struct pminmax_impl<PropagateNumbers>
    {
        template <typename Packet, typename Op> static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a, const Packet& b, Op op)
        {
            Packet not_nan_mask_a = pcmp_eq(a, a);
            Packet not_nan_mask_b = pcmp_eq(b, b);
            return pselect(not_nan_mask_a, pselect(not_nan_mask_b, op(a, b), a), b);
        }
    };

#ifndef SYCL_DEVICE_ONLY
#define EIGEN_BINARY_OP_NAN_PROPAGATION(Type, Func) Func
#else
#define EIGEN_BINARY_OP_NAN_PROPAGATION(Type, Func) [](const Type& a, const Type& b) { return Func(a, b); }
#endif

    /** \internal \returns the min of \a a and \a b  (coeff-wise).
    If \a a or \b b is NaN, the return value is implementation defined. */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pmin(const Packet& a, const Packet& b) { return numext::mini(a, b); }

    /** \internal \returns the min of \a a and \a b  (coeff-wise).
    NaNPropagation determines the NaN propagation semantics. */
    template <int NaNPropagation, typename Packet> EIGEN_DEVICE_FUNC inline Packet pmin(const Packet& a, const Packet& b)
    {
        return pminmax_impl<NaNPropagation>::run(a, b, EIGEN_BINARY_OP_NAN_PROPAGATION(Packet, (pmin<Packet>)));
    }

    /** \internal \returns the max of \a a and \a b  (coeff-wise)
    If \a a or \b b is NaN, the return value is implementation defined. */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pmax(const Packet& a, const Packet& b) { return numext::maxi(a, b); }

    /** \internal \returns the max of \a a and \a b  (coeff-wise).
    NaNPropagation determines the NaN propagation semantics. */
    template <int NaNPropagation, typename Packet> EIGEN_DEVICE_FUNC inline Packet pmax(const Packet& a, const Packet& b)
    {
        return pminmax_impl<NaNPropagation>::run(a, b, EIGEN_BINARY_OP_NAN_PROPAGATION(Packet, (pmax<Packet>)));
    }

    /** \internal \returns the absolute value of \a a */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pabs(const Packet& a) { return numext::abs(a); }
    template <> EIGEN_DEVICE_FUNC inline unsigned int pabs(const unsigned int& a) { return a; }
    template <> EIGEN_DEVICE_FUNC inline unsigned long pabs(const unsigned long& a) { return a; }
    template <> EIGEN_DEVICE_FUNC inline unsigned long long pabs(const unsigned long long& a) { return a; }

    /** \internal \returns the addsub value of \a a,b */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet paddsub(const Packet& a, const Packet& b)
    {
        return pselect(peven_mask(a), padd(a, b), psub(a, b));
    }

    /** \internal \returns the phase angle of \a a */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet parg(const Packet& a)
    {
        using numext::arg;
        return arg(a);
    }

    /** \internal \returns \a a logically shifted by N bits to the right */
    template <int N> EIGEN_DEVICE_FUNC inline int parithmetic_shift_right(const int& a) { return a >> N; }
    template <int N> EIGEN_DEVICE_FUNC inline long int parithmetic_shift_right(const long int& a) { return a >> N; }

    /** \internal \returns \a a arithmetically shifted by N bits to the right */
    template <int N> EIGEN_DEVICE_FUNC inline int plogical_shift_right(const int& a) { return static_cast<int>(static_cast<unsigned int>(a) >> N); }
    template <int N> EIGEN_DEVICE_FUNC inline long int plogical_shift_right(const long int& a) { return static_cast<long>(static_cast<unsigned long>(a) >> N); }

    /** \internal \returns \a a shifted by N bits to the left */
    template <int N> EIGEN_DEVICE_FUNC inline int plogical_shift_left(const int& a) { return a << N; }
    template <int N> EIGEN_DEVICE_FUNC inline long int plogical_shift_left(const long int& a) { return a << N; }

    /** \internal \returns the significant and exponent of the underlying floating point numbers
  * See https://en.cppreference.com/w/cpp/numeric/math/frexp
  */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pfrexp(const Packet& a, Packet& exponent)
    {
        int exp;
        EIGEN_USING_STD(frexp);
        Packet result = static_cast<Packet>(frexp(a, &exp));
        exponent = static_cast<Packet>(exp);
        return result;
    }

    /** \internal \returns a * 2^((int)exponent)
  * See https://en.cppreference.com/w/cpp/numeric/math/ldexp
  */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pldexp(const Packet& a, const Packet& exponent)
    {
        EIGEN_USING_STD(ldexp)
        return static_cast<Packet>(ldexp(a, static_cast<int>(exponent)));
    }

    /** \internal \returns the min of \a a and \a b  (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pabsdiff(const Packet& a, const Packet& b)
    {
        return pselect(pcmp_lt(a, b), psub(b, a), psub(a, b));
    }

    /** \internal \returns a packet version of \a *from, from must be 16 bytes aligned */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pload(const typename unpacket_traits<Packet>::type* from) { return *from; }

    /** \internal \returns a packet version of \a *from, (un-aligned load) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet ploadu(const typename unpacket_traits<Packet>::type* from) { return *from; }

    /** \internal \returns a packet version of \a *from, (un-aligned masked load)
 * There is no generic implementation. We only have implementations for specialized
 * cases. Generic case should not be called.
 */
    template <typename Packet>
    EIGEN_DEVICE_FUNC inline typename enable_if<unpacket_traits<Packet>::masked_load_available, Packet>::type
    ploadu(const typename unpacket_traits<Packet>::type* from, typename unpacket_traits<Packet>::mask_t umask);

    /** \internal \returns a packet with constant coefficients \a a, e.g.: (a,a,a,a) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pset1(const typename unpacket_traits<Packet>::type& a) { return a; }

    /** \internal \returns a packet with constant coefficients set from bits */
    template <typename Packet, typename BitsType> EIGEN_DEVICE_FUNC inline Packet pset1frombits(BitsType a);

    /** \internal \returns a packet with constant coefficients \a a[0], e.g.: (a[0],a[0],a[0],a[0]) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pload1(const typename unpacket_traits<Packet>::type* a) { return pset1<Packet>(*a); }

    /** \internal \returns a packet with elements of \a *from duplicated.
  * For instance, for a packet of 8 elements, 4 scalars will be read from \a *from and
  * duplicated to form: {from[0],from[0],from[1],from[1],from[2],from[2],from[3],from[3]}
  * Currently, this function is only used for scalar * complex products.
  */
    template <typename Packet> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet ploaddup(const typename unpacket_traits<Packet>::type* from) { return *from; }

    /** \internal \returns a packet with elements of \a *from quadrupled.
  * For instance, for a packet of 8 elements, 2 scalars will be read from \a *from and
  * replicated to form: {from[0],from[0],from[0],from[0],from[1],from[1],from[1],from[1]}
  * Currently, this function is only used in matrix products.
  * For packet-size smaller or equal to 4, this function is equivalent to pload1
  */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet ploadquad(const typename unpacket_traits<Packet>::type* from) { return pload1<Packet>(from); }

    /** \internal equivalent to
  * \code
  * a0 = pload1(a+0);
  * a1 = pload1(a+1);
  * a2 = pload1(a+2);
  * a3 = pload1(a+3);
  * \endcode
  * \sa pset1, pload1, ploaddup, pbroadcast2
  */
    template <typename Packet>
    EIGEN_DEVICE_FUNC inline void pbroadcast4(const typename unpacket_traits<Packet>::type* a, Packet& a0, Packet& a1, Packet& a2, Packet& a3)
    {
        a0 = pload1<Packet>(a + 0);
        a1 = pload1<Packet>(a + 1);
        a2 = pload1<Packet>(a + 2);
        a3 = pload1<Packet>(a + 3);
    }

    /** \internal equivalent to
  * \code
  * a0 = pload1(a+0);
  * a1 = pload1(a+1);
  * \endcode
  * \sa pset1, pload1, ploaddup, pbroadcast4
  */
    template <typename Packet> EIGEN_DEVICE_FUNC inline void pbroadcast2(const typename unpacket_traits<Packet>::type* a, Packet& a0, Packet& a1)
    {
        a0 = pload1<Packet>(a + 0);
        a1 = pload1<Packet>(a + 1);
    }

    /** \internal \brief Returns a packet with coefficients (a,a+1,...,a+packet_size-1). */
    template <typename Packet> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet plset(const typename unpacket_traits<Packet>::type& a) { return a; }

    /** \internal \returns a packet with constant coefficients \a a, e.g.: (x, 0, x, 0),
     where x is the value of all 1-bits. */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet peven_mask(const Packet& /*a*/)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        const size_t n = unpacket_traits<Packet>::size;
        EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) Scalar elements[n];
        for (size_t i = 0; i < n; ++i) { memset(elements + i, ((i & 1) == 0 ? 0xff : 0), sizeof(Scalar)); }
        return ploadu<Packet>(elements);
    }

    /** \internal copy the packet \a from to \a *to, \a to must be 16 bytes aligned */
    template <typename Scalar, typename Packet> EIGEN_DEVICE_FUNC inline void pstore(Scalar* to, const Packet& from) { (*to) = from; }

    /** \internal copy the packet \a from to \a *to, (un-aligned store) */
    template <typename Scalar, typename Packet> EIGEN_DEVICE_FUNC inline void pstoreu(Scalar* to, const Packet& from) { (*to) = from; }

    /** \internal copy the packet \a from to \a *to, (un-aligned store with a mask)
 * There is no generic implementation. We only have implementations for specialized
 * cases. Generic case should not be called.
 */
    template <typename Scalar, typename Packet>
    EIGEN_DEVICE_FUNC inline typename enable_if<unpacket_traits<Packet>::masked_store_available, void>::type
    pstoreu(Scalar* to, const Packet& from, typename unpacket_traits<Packet>::mask_t umask);

    template <typename Scalar, typename Packet> EIGEN_DEVICE_FUNC inline Packet pgather(const Scalar* from, Index /*stride*/) { return ploadu<Packet>(from); }

    template <typename Scalar, typename Packet> EIGEN_DEVICE_FUNC inline void pscatter(Scalar* to, const Packet& from, Index /*stride*/) { pstore(to, from); }

    /** \internal tries to do cache prefetching of \a addr */
    template <typename Scalar> EIGEN_DEVICE_FUNC inline void prefetch(const Scalar* addr)
    {
#if defined(EIGEN_HIP_DEVICE_COMPILE)
        // do nothing
#elif defined(EIGEN_CUDA_ARCH)
#if defined(__LP64__) || EIGEN_OS_WIN64
        // 64-bit pointer operand constraint for inlined asm
        asm(" prefetch.L1 [ %1 ];" : "=l"(addr) : "l"(addr));
#else
        // 32-bit pointer operand constraint for inlined asm
        asm(" prefetch.L1 [ %1 ];" : "=r"(addr) : "r"(addr));
#endif
#elif (!EIGEN_COMP_MSVC) && (EIGEN_COMP_GNUC || EIGEN_COMP_CLANG || EIGEN_COMP_ICC)
        __builtin_prefetch(addr);
#endif
    }

    /** \internal \returns the reversed elements of \a a*/
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet preverse(const Packet& a) { return a; }

    /** \internal \returns \a a with real and imaginary part flipped (for complex type only) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pcplxflip(const Packet& a) { return Packet(numext::imag(a), numext::real(a)); }

    /**************************
* Special math functions
***************************/

    /** \internal \returns the sine of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psin(const Packet& a)
    {
        EIGEN_USING_STD(sin);
        return sin(a);
    }

    /** \internal \returns the cosine of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcos(const Packet& a)
    {
        EIGEN_USING_STD(cos);
        return cos(a);
    }

    /** \internal \returns the tan of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet ptan(const Packet& a)
    {
        EIGEN_USING_STD(tan);
        return tan(a);
    }

    /** \internal \returns the arc sine of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pasin(const Packet& a)
    {
        EIGEN_USING_STD(asin);
        return asin(a);
    }

    /** \internal \returns the arc cosine of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pacos(const Packet& a)
    {
        EIGEN_USING_STD(acos);
        return acos(a);
    }

    /** \internal \returns the arc tangent of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patan(const Packet& a)
    {
        EIGEN_USING_STD(atan);
        return atan(a);
    }

    /** \internal \returns the hyperbolic sine of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psinh(const Packet& a)
    {
        EIGEN_USING_STD(sinh);
        return sinh(a);
    }

    /** \internal \returns the hyperbolic cosine of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcosh(const Packet& a)
    {
        EIGEN_USING_STD(cosh);
        return cosh(a);
    }

    /** \internal \returns the hyperbolic tan of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet ptanh(const Packet& a)
    {
        EIGEN_USING_STD(tanh);
        return tanh(a);
    }

    /** \internal \returns the exp of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexp(const Packet& a)
    {
        EIGEN_USING_STD(exp);
        return exp(a);
    }

    /** \internal \returns the expm1 of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexpm1(const Packet& a) { return numext::expm1(a); }

    /** \internal \returns the log of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog(const Packet& a)
    {
        EIGEN_USING_STD(log);
        return log(a);
    }

    /** \internal \returns the log1p of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog1p(const Packet& a) { return numext::log1p(a); }

    /** \internal \returns the log10 of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog10(const Packet& a)
    {
        EIGEN_USING_STD(log10);
        return log10(a);
    }

    /** \internal \returns the log10 of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog2(const Packet& a)
    {
        typedef typename internal::unpacket_traits<Packet>::type Scalar;
        return pmul(pset1<Packet>(Scalar(EIGEN_LOG2E)), plog(a));
    }

    /** \internal \returns the square-root of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psqrt(const Packet& a) { return numext::sqrt(a); }

    /** \internal \returns the reciprocal square-root of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet prsqrt(const Packet& a)
    {
        typedef typename internal::unpacket_traits<Packet>::type Scalar;
        return pdiv(pset1<Packet>(Scalar(1)), psqrt(a));
    }

    /** \internal \returns the rounded value of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pround(const Packet& a)
    {
        using numext::round;
        return round(a);
    }

    /** \internal \returns the floor of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pfloor(const Packet& a)
    {
        using numext::floor;
        return floor(a);
    }

    /** \internal \returns the rounded value of \a a (coeff-wise) with current
 * rounding mode */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet print(const Packet& a)
    {
        using numext::rint;
        return rint(a);
    }

    /** \internal \returns the ceil of \a a (coeff-wise) */
    template <typename Packet> EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pceil(const Packet& a)
    {
        using numext::ceil;
        return ceil(a);
    }

    /** \internal \returns the first element of a packet */
    template <typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type pfirst(const Packet& a) { return a; }

    /** \internal \returns the sum of the elements of upper and lower half of \a a if \a a is larger than 4.
  * For a packet {a0, a1, a2, a3, a4, a5, a6, a7}, it returns a half packet {a0+a4, a1+a5, a2+a6, a3+a7}
  * For packet-size smaller or equal to 4, this boils down to a noop.
  */
    template <typename Packet>
    EIGEN_DEVICE_FUNC inline typename conditional<(unpacket_traits<Packet>::size % 8) == 0, typename unpacket_traits<Packet>::half, Packet>::type
    predux_half_dowto4(const Packet& a)
    {
        return a;
    }

    // Slow generic implementation of Packet reduction.
    template <typename Packet, typename Op> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_helper(const Packet& a, Op op)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        const size_t n = unpacket_traits<Packet>::size;
        EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) Scalar elements[n];
        pstoreu<Scalar>(elements, a);
        for (size_t k = n / 2; k > 0; k /= 2)
        {
            for (size_t i = 0; i < k; ++i) { elements[i] = op(elements[i], elements[i + k]); }
        }
        return elements[0];
    }

    /** \internal \returns the sum of the elements of \a a*/
    template <typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux(const Packet& a) { return a; }

    /** \internal \returns the product of the elements of \a a */
    template <typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_mul(const Packet& a)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmul<Scalar>)));
    }

    /** \internal \returns the min of the elements of \a a */
    template <typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_min(const Packet& a)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmin<PropagateFast, Scalar>)));
    }

    template <int NaNPropagation, typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_min(const Packet& a)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmin<NaNPropagation, Scalar>)));
    }

    /** \internal \returns the min of the elements of \a a */
    template <typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_max(const Packet& a)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmax<PropagateFast, Scalar>)));
    }

    template <int NaNPropagation, typename Packet> EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_max(const Packet& a)
    {
        typedef typename unpacket_traits<Packet>::type Scalar;
        return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmax<NaNPropagation, Scalar>)));
    }

#undef EIGEN_BINARY_OP_NAN_PROPAGATION

    /** \internal \returns true if all coeffs of \a a means "true"
  * It is supposed to be called on values returned by pcmp_*.
  */
    // not needed yet
    // template<typename Packet> EIGEN_DEVICE_FUNC inline bool predux_all(const Packet& a)
    // { return bool(a); }

    /** \internal \returns true if any coeffs of \a a means "true"
  * It is supposed to be called on values returned by pcmp_*.
  */
    template <typename Packet> EIGEN_DEVICE_FUNC inline bool predux_any(const Packet& a)
    {
        // Dirty but generic implementation where "true" is assumed to be non 0 and all the sames.
        // It is expected that "true" is either:
        //  - Scalar(1)
        //  - bits full of ones (NaN for floats),
        //  - or first bit equals to 1 (1 for ints, smallest denormal for floats).
        // For all these cases, taking the sum is just fine, and this boils down to a no-op for scalars.
        typedef typename unpacket_traits<Packet>::type Scalar;
        return numext::not_equal_strict(predux(a), Scalar(0));
    }

    /***************************************************************************
* The following functions might not have to be overwritten for vectorized types
***************************************************************************/

    /** \internal copy a packet with constant coefficient \a a (e.g., [a,a,a,a]) to \a *to. \a to must be 16 bytes aligned */
    // NOTE: this function must really be templated on the packet type (think about different packet types for the same scalar type)
    template <typename Packet> inline void pstore1(typename unpacket_traits<Packet>::type* to, const typename unpacket_traits<Packet>::type& a)
    {
        pstore(to, pset1<Packet>(a));
    }

    /** \internal \returns a * b + c (coeff-wise) */
    template <typename Packet> EIGEN_DEVICE_FUNC inline Packet pmadd(const Packet& a, const Packet& b, const Packet& c) { return padd(pmul(a, b), c); }

    /** \internal \returns a packet version of \a *from.
  * The pointer \a from must be aligned on a \a Alignment bytes boundary. */
    template <typename Packet, int Alignment> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet ploadt(const typename unpacket_traits<Packet>::type* from)
    {
        if (Alignment >= unpacket_traits<Packet>::alignment)
            return pload<Packet>(from);
        else
            return ploadu<Packet>(from);
    }

    /** \internal copy the packet \a from to \a *to.
  * The pointer \a from must be aligned on a \a Alignment bytes boundary. */
    template <typename Scalar, typename Packet, int Alignment> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE void pstoret(Scalar* to, const Packet& from)
    {
        if (Alignment >= unpacket_traits<Packet>::alignment)
            pstore(to, from);
        else
            pstoreu(to, from);
    }

    /** \internal \returns a packet version of \a *from.
  * Unlike ploadt, ploadt_ro takes advantage of the read-only memory path on the
  * hardware if available to speedup the loading of data that won't be modified
  * by the current computation.
  */
    template <typename Packet, int LoadMode> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet ploadt_ro(const typename unpacket_traits<Packet>::type* from)
    {
        return ploadt<Packet, LoadMode>(from);
    }

/***************************************************************************
* Fast complex products (GCC generates a function call which is very slow)
***************************************************************************/

// Eigen+CUDA does not support complexes.
#if !defined(EIGEN_GPUCC)

    template <> inline std::complex<float> pmul(const std::complex<float>& a, const std::complex<float>& b)
    {
        return std::complex<float>(a.real() * b.real() - a.imag() * b.imag(), a.imag() * b.real() + a.real() * b.imag());
    }

    template <> inline std::complex<double> pmul(const std::complex<double>& a, const std::complex<double>& b)
    {
        return std::complex<double>(a.real() * b.real() - a.imag() * b.imag(), a.imag() * b.real() + a.real() * b.imag());
    }

#endif

    /***************************************************************************
 * PacketBlock, that is a collection of N packets where the number of words
 * in the packet is a multiple of N.
***************************************************************************/
    template <typename Packet, int N = unpacket_traits<Packet>::size> struct PacketBlock
    {
        Packet packet[N];
    };

    template <typename Packet> EIGEN_DEVICE_FUNC inline void ptranspose(PacketBlock<Packet, 1>& /*kernel*/)
    {
        // Nothing to do in the scalar case, i.e. a 1x1 matrix.
    }

    /***************************************************************************
 * Selector, i.e. vector of N boolean values used to select (i.e. blend)
 * words from 2 packets.
***************************************************************************/
    template <size_t N> struct Selector
    {
        bool select[N];
    };

    template <typename Packet>
    EIGEN_DEVICE_FUNC inline Packet pblend(const Selector<unpacket_traits<Packet>::size>& ifPacket, const Packet& thenPacket, const Packet& elsePacket)
    {
        return ifPacket.select[0] ? thenPacket : elsePacket;
    }

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_GENERIC_PACKET_MATH_H
